Competition - post a photo you've taken of nature >>

How do you find the cartesian equation of a curve? (C4)

Example question:
A curve is define by the parametric equations x=(t^2/2) + 1 and y=(4/t) -1
When t= 2, gradient is dy/dx = -1/2
Find a cartesian equation of the curve.

We havent learnt how to do it in school. I got this question from a past paper, and was just wondering how you'd do it.
Thanks!

Reply 1

ODES_PDES

Original post by Phoebus Apollo

Solve the easier equation for t and substitute into the other.

Reply 2

username1560589

Or remove t from the simultaneous equations by other means eg. dividing.

Reply 3

Zacken

Original post by Phoebus Apollo

It's kind of like solving simultaneous equations except trickier. Basically you want to see what you can do with

x

and

y

to eliminate

t

, that'll get you

x

in terms of

y

y

in terms of

x

as required.

Here, you can kind of brute force it by doing

y + 1 = \frac{4}{t} \Rightarrow t = \frac{4}{y+1}

x = \frac{\frac{16}{(y+1)^2}}{2} + 1

.

But generally, you'll need to think about eliminating t and when you have trig parametric equations, you'll need to think about trigonometrical identities, etc...

Reply 4

B_9710

For this type. Find t in terms of x and then substitute this expression of t into the equation for y. This eliminates t from the y equation leaving an equation in terms of x and y only - the Cartesian equation.

Reply 5

nerak99

Original post by Zacken

...

Here, you can kind of brute force it by doing

y + 1 = \frac{4}{t} \Rightarrow t = \frac{4}{y+1}

x = \frac{\frac{16}{(y+1)^2}}{2} + 1

.

But generally, you'll need to think about eliminating t and when you have trig parametric equations, you'll need to think about trigonometrical identities, etc...

and you would complete with

x = \frac{\frac{16}{(y+1)^2}}{2} + 1 = \frac{32}{(y+1)^2}+1

.<<<< The 32 should be 8 but leaving it for the sake of history

Because you would never get a mark at C4 with

x = \frac{\frac{16}{(y+1)^2}}{2} + 1

.

Zacken knows this but you might not

(edited 7 years ago)

Reply 6

Phoebus Apollo

Original post by ODES_PDES

Solve the easier equation for t and substitute into the other.

Original post by morgan8002

Or remove t from the simultaneous equations by other means eg. dividing.

Original post by Zacken

It's kind of like solving simultaneous equations except trickier. Basically you want to see what you can do with

x

and

y

to eliminate

t

, that'll get you

x

in terms of

y

y

in terms of

x

as required.

Here, you can kind of brute force it by doing

y + 1 = \frac{4}{t} \Rightarrow t = \frac{4}{y+1}

x = \frac{\frac{16}{(y+1)^2}}{2} + 1

.

But generally, you'll need to think about eliminating t and when you have trig parametric equations, you'll need to think about trigonometrical identities, etc...

Original post by B_9710

Original post by nerak99

and you would complete with

x = \frac{\frac{16}{(y+1)^2}}{2} + 1 = \frac{32}{(y+1)^2}+1

.

Because you would never get a mark at C4 with

x = \frac{\frac{16}{(y+1)^2}}{2} + 1

.

Zacken knows this but you might not

Thanks everyone :smile:

Reply 7

Zacken

Original post by nerak99

and you would complete with

x = \frac{\frac{16}{(y+1)^2}}{2} + 1 = \frac{32}{(y+1)^2}+1

Surely it'd be

x = \frac{8}{(y+1)^2} + 1

Reply 8

nerak99

Original post by Zacken

Surely it'd be

x = \frac{8}{(y+1)^2} + 1

Well I am nervous of arguing with you Zacken and you are correct

a) You kind of make my point and

b) You are right because I read it as a/(b/c) instead of (a/b)/c

Reply 9

username2176541

Original post by Zacken

Surely it'd be

x = \frac{8}{(y+1)^2} + 1

Brackets would help for clarity.

Reply 10

Zacken

Original post by Xenon17

Brackets would help for clarity.

How would it?

Reply 11

nerak99

Original post by Zacken

How would it?

. Could be written

\frac{\left(\frac{16}{(y+1)^2} \right)}{2} +1

(which is unambiguous)

Although it still is a massive carbuncle.

(edited 7 years ago)

Reply 12

Zacken

Original post by nerak99

. Could be written

\frac{\left(\frac{16}{(y+1)^2} \right)}{2} +1

That's ruins the aesthetic and doesn't really help clear up any ambiguity since there is a visible and marked difference between

\displaystyle x = \frac{\frac{16}{(y+1)^2}}{2}

and

\displaystyle x = \frac{16}{\frac{(y+1)^2}{2}}

Reply 13

nerak99

Original post by Zacken

That's ruins the aesthetic and doesn't really help clear up any ambiguity since there is a visible and marked difference between

\displaystyle x = \frac{\frac{16}{(y+1)^2}}{2}

and

\displaystyle x = \frac{16}{\frac{(y+1)^2}{2}}

Well There is no aesthetic in the double fraction form anyway but at least sticking brackets around the toip fraction makes it clear. If this were written by hand it would look like it could be a/b/c or a/b/c (ironic face) Hence the (a/b)/c we have here should be resolved to a/(bc) (smiley face, gritted teeth face, I am going for my tea now face)